Canteen Survey, Independence and a Supplier Minimum
A college canteen in Mangaluru surveyed 400 students about their preferred drink. Of the 240 boys, 120 chose lime juice and the rest chose buttermilk. Of the 160 girls, 40 chose lime juice and the rest chose buttermilk. The canteen manager plans to order 200 bottles of lime juice and 200 bottles of buttermilk for next week, assuming drink preference is independent of gender. A supplier offers a discount only if the lime-juice order is at least 220 bottles. Next week the college hosts an inter-college tournament, so the canteen expects 500 students, 70% of them boys. The manager wants to know whether the independence assumption is supported by the survey and whether the 220-bottle minimum should be accepted. Assume every student buys exactly one bottle and the survey proportions apply to next week's students except for the stated gender mix.
1. Compute P(lime | boy), P(lime | girl) and P(lime). Are gender and drink choice independent? Show the test.
Read the worked answer
P(lime | boy) = 120/240 = 0.50. P(lime | girl) = 40/160 = 0.25. P(lime) = (120 + 40)/400 = 160/400 = 0.40.
For independence, the conditional probabilities should equal the marginal probability. Here 0.50 and 0.25 are not equal to each other or to 0.40, so gender and choice are not independent.
Equivalently, P(lime and boy) = 120/400 = 0.30, while P(lime)P(boy) = 0.40 × 0.60 = 0.24. Since 0.30 ≠ 0.24, the events are dependent. Boys are relatively more likely to choose lime juice.
2. Using the survey's conditional probabilities, estimate lime-juice demand for next week's 500 students if 70% are boys. Should the canteen accept the 220-bottle minimum?
Read the worked answer
Next week: boys = 0.70 × 500 = 350; girls = 150. Expected lime demand = 350 × 0.50 + 150 × 0.25 = 175 + 37.5 = 212.5, about 213 bottles.
A minimum order of 220 bottles is about 7 bottles above expected demand. Accepting it creates expected excess stock of roughly 7 bottles. The decision should compare the discount value with the loss on those extra bottles and the risk that demand is lower than expected. If the discount is small, order about 213 bottles and forgo the minimum; if the discount comfortably exceeds the expected loss, accepting 220 can be justified.
3. A student is selected at random from the 400 and is found to prefer lime juice. What is the probability the student is a boy? Does this support the manager's plan to stock equal numbers?
Read the worked answer
Among the 160 lime-juice choosers, 120 are boys. So P(boy | lime) = 120/160 = 0.75.
This supports the survey's dependence finding: lime-juice buyers are predominantly boys. The manager's equal 200/200 plan ignores gender composition. With next week's expected 500 students, the estimate is about 213 lime and 287 buttermilk, not 200 each. Equal stocking would likely understock buttermilk and overstock lime. The manager should use the conditional probabilities and the expected gender mix, not an independence assumption.